Question: $ F = \left[\begin{array}{rrr}0 & -1 & 0 \\ 2 & 5 & 0\end{array}\right]$ $ E = \left[\begin{array}{rr}1 & 1 \\ 3 & 3 \\ 3 & -2\end{array}\right]$ What is $ F E$ ?
Explanation: Because $ F$ has dimensions $(2\times3)$ and $ E$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ F E = \left[\begin{array}{rrr}{0} & {-1} & {0} \\ {2} & {5} & {0}\end{array}\right] \left[\begin{array}{rr}{1} & \color{#DF0030}{1} \\ {3} & \color{#DF0030}{3} \\ {3} & \color{#DF0030}{-2}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ F$ , with the corresponding elements in column $j$ of the second matrix, $ E$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ F$ with the first element in ${\text{column }1}$ of $ E$ , then multiply the second element in ${\text{row }1}$ of $ F$ with the second element in ${\text{column }1}$ of $ E$ , and so on. Add the products together. $ \left[\begin{array}{rr}{0}\cdot{1}+{-1}\cdot{3}+{0}\cdot{3} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ F$ with the corresponding elements in ${\text{column }1}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{0}\cdot{1}+{-1}\cdot{3}+{0}\cdot{3} & ? \\ {2}\cdot{1}+{5}\cdot{3}+{0}\cdot{3} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ F$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{0}\cdot{1}+{-1}\cdot{3}+{0}\cdot{3} & {0}\cdot\color{#DF0030}{1}+{-1}\cdot\color{#DF0030}{3}+{0}\cdot\color{#DF0030}{-2} \\ {2}\cdot{1}+{5}\cdot{3}+{0}\cdot{3} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{0}\cdot{1}+{-1}\cdot{3}+{0}\cdot{3} & {0}\cdot\color{#DF0030}{1}+{-1}\cdot\color{#DF0030}{3}+{0}\cdot\color{#DF0030}{-2} \\ {2}\cdot{1}+{5}\cdot{3}+{0}\cdot{3} & {2}\cdot\color{#DF0030}{1}+{5}\cdot\color{#DF0030}{3}+{0}\cdot\color{#DF0030}{-2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}-3 & -3 \\ 17 & 17\end{array}\right] $